[1X3 A description of the Lie algebras that are contained in the package[0X
[1X3.1 Description of the non-solvable Lie algebras[0X
In this section we list the non-solvable Lie algebras contained in the
package. Our notation follows [Str], where a more detailed description can
also be found. In particular if $L$ is a Lie algebra over $F$ then $C(L)$
denotes the center of $L$. Further, if $x_1,\ldots,x_k$ are elements of $L$,
then $F<x_1,\ldots,x_k>$ denotes the linear subspace generated by
$x_1,\ldots,x_k$, and we also write $Fx_1$ for $F<x_1>$
[1X3.2 Dimension 3[0X
There are no non-solvable Lie algebras with dimension 1 or 2. Over an
arbitrary finite field [3XF[0m, there is just one isomorphism type of non-solvable
Lie algebras:
(1) If [3Xchar F=2[0m then the algebra is $W(1;\underline 2)^{(1)}$.
(2) If [3Xchar F>2[0m then the algebra is $\mbox{sl}(2,F)$.
See Theorem 3.2 of [Str] for details.
[1X3.3 Dimension 4[0X
Over a finite field [3XF[0m of characteristic 2 there are two isomorphism classes
of non-solvable Lie algebras with dimension 4, while over a finite field [3XF[0m
of odd characteristic the number of isomorphism classes is one (see Theorem
4.1 of [Str]). The classes are as follows:
(1) characteristic 2: $W(1;\underline 2)$ and $W(1;\underline
2)^{(1)}\oplus F$.
(2) odd characteristic: $\mbox{gl}(2,F)$.
[1X3.4 Dimension 5[0X
[1X3.4-1 Characteristic 2[0X
Over a finite field [3XF[0m of characteristic 2 there are 5 isomorphism classes of
non-solvable Lie algebras with dimension 5:
(1) $\mbox{Der}(W(1;\underline 2)^{(1)})$;
(2) $W(1;\underline 2)\ltimes Fu$ where $[W(1;\underline 2)^{(1)},u]=0$,
$[x^{(3)}\partial,u]=\delta u$ and $\delta\in\{0,1\}$ (two algebras);
(3) $W(1;\underline 2)^{(1)}\oplus(F\left< h,u\right>)$, $[h,u]=\delta u$,
where $\delta\in\{0,1\}$ (two algebras).
See Theorem 4.2 of [Str] for details.
[1X3.4-2 Odd characteristic[0X
Over a field $F$of odd characteristic the number of isomorphism types of
5-dimensional non-solvable Lie algebras is $3$ if the characteristic is at
least 7, and it is 4 otherwise (see Theorem 4.3 of [Str]). The classes are
as follows.
(1) $\mbox{sl}(2,F)\oplus F<x,y>$, $[x,y]=\delta y$ where
$\delta\in\{0,1\}$.
(2) $\mbox{sl}(2,F)\ltimes V(1)$ where $V(1)$ is the irreducible
2-dimensional $\mbox{sl}(2,F)$-module.
(3) If $\mbox{char }F=3$ then there is an additional algebra, namely the
non-split extension $0\rightarrow V(1)\rightarrow
L\rightarrow\mbox{sl}(2,F)\rightarrow 0$.
(4) If $\mbox{char }F=5$ then there is an additional algebra:
$W(1;\underline 1)$.
[1X3.5 Dimension 6[0X
[1X3.5-1 Characteristic 2[0X
Over a field $F$ of characteristic 2, the isomorphism classes of
non-solvable Lie algebras are as follows.
(1) $W(1;\underline 2)^{(1)}\oplus W(1;\underline 2)^{(1)}$.
(2) $W(1;\underline 2)^{(1)}\otimes F_{q^2}$ where $F=F_q$.
(3) $\mbox{Der}(W(1;\underline 2)^{(1)})\ltimes Fu$, $[W(1;\underline
2),u]=0$, $[\partial^2,u]=\delta u$ where $\delta=\{0,1\}$.
(4) $W(1;\underline 2)\ltimes (F<h,u>)$, $[W(1;\underline
2)^{(1)},(F<h,u>]=0$, $[h,u]=\delta u$, and if $\delta=0$, then the
action of $x^{(3)}\partial$ on $F<h,u>$ is given by one of the
following matrices:
\left(\begin{array}{cc} 0 & 0\\ 0 & 0\end{array}\right),\
\left(\begin{array}{cc} 0 & 1\\ 0 & 0\end{array}\right),\
\left(\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right),\
\left(\begin{array}{cc} 1 & 1\\ 0 & 1\end{array}\right),\
\left(\begin{array}{cc} 0 & \xi\\ 1 &
1\end{array}\right)\mbox{ where }\xi\in F^*.
(5) the algebra is as in (4.), but $\delta=1$. Note that Theorem 5.1(3/b)
of [Str] lists two such algebras but they turn out to be isomorphic.
We take the one with $[x^{(3)}\partial,h]=[x^{(3)}\partial,u]=0$.
(6) $W(1;\underline 2)^{(1)}\oplus K$ where $K$ is a 3-dimensional
solvable Lie algebra.
(7) $W(1;\underline 2)^{(1)}\ltimes \mathcal O(1;\underline 2)/F$.
(8) the non-split extension $0\rightarrow \mathcal O(1;\underline
2)/F\rightarrow L\rightarrow W(1;\underline 2)^{(1)}\rightarrow 0$.
See Theorem 5.1 of [Str].
[1X3.5-2 General odd characteristic[0X
If the characteristic of the field is odd, then the 6-dimensional
non-solvable Lie algebras are described by Theorems 5.2--5.4 of [Str]. Over
such a field $F$, let us define the following isomorphism classes of
6-dimensional non-solvable Lie algebras.
(1) $\mbox{sl}(2,F)\oplus\mbox{sl}(2,F) $.
(2) $\mbox{sl}(2,F_{q^2})$ where $F=F_q$;
(3) $\mbox{sl}(2,F)\oplus K$ where $K$ is a solvable Lie algebra with
dimension 3;
(4) $\mbox{sl}(2,F)\ltimes (V(0)\oplus V(1))$ where $V(i)$ is the
$(i+1)$-dimensional irreducible $\mbox{sl}(2,F)$-module;
(5) $\mbox{sl}(2,F)\ltimes V(2)$ where $V(2)$ is the $3$-dimensional
irreducible $\mbox{sl}(2,F)$-module;
(6) $\mbox{sl}(2,F)\ltimes(V(1)\oplus C(L))\cong \mbox{sl}(2,F)\ltimes H$
where $H$ is the Heisenberg Lie algebra;
(7) $\mbox{sl}(2,F)\ltimes K$ where $K=Fd\oplus K^{(1)}$, $K^{(1)}$ is
2-dimensional abelian, isomorphic, as an $\mbox{sl}(2,F)$-module, to
$V(1)$, $[\mbox{sl}(2,F),d]=0$, and, for all $v\in K$, $[d,v]=v$;
If the characteristic of $F$ is at least 7, then these algebras form a
complete and irredundant list of the isomorphism classes of the
6-dimensional non-solvable Lie algebras.
[1X3.5-3 Characteristic 3[0X
If the characteristic of the field $F$ is 3, then, besides the classes in
Section [14X3.5-2[0m, we also obtain the following isomorphism classes.
(1) $\mbox{sl}(2,F)\ltimes V(2,\chi)$ where $\chi$ is a 3-dimensional
character of $\mbox{sl}(2,F)$. Each such character is described by a
field element $\xi$ such that $T^3+T^2-\xi$ has a root in $F$; see
Proposition 3.5 of [Str] for more details.
(2) $W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)$ where $\mathcal
O(1;\underline 1)$ is considered as an abelian Lie algebra.
(3) $W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)^*$ where $\mathcal
O(1;\underline 1)^*$ is the dual of $\mathcal O(1;\underline 1)$ and
it is considered as an abelian Lie algebra.
(4) One of the two 6-dimensional central extensions of the non-split
extension $0\rightarrow V(1)\rightarrow L\rightarrow
\mbox{sl}(2,F)\rightarrow 0$; see Proposition 4.5 of [Str]. We note
that Proposition 4.5 of [Str] lists three such central extensions, but
one of them is not a Lie algebra.
(5) One of the two non-split extensions $0\rightarrow\mbox{rad }
L\rightarrow L\rightarrow L/\mbox{rad } L\rightarrow 0$ with a
5-dimensional ideal; see Theorem 5.4 of [Str].
We note here that [Str] lists one more non-solvable Lie algebra over a field
of characteristic 3, namely the one in Theorem 5.3(5). However, this algebra
is isomorphic to the one in Theorem 5.3(4).
[1X3.5-4 Characteristic 5[0X
If the characteristic of the field $F$ is 5, then, besides the classes in
Section [14X3.5-2[0m, we also obtain the following isomorphism classes.
(1) $W(1;\underline 1)\oplus F$.
(2) The non-split central extension $0\rightarrow F\rightarrow
L\rightarrow W(1;\underline 1)\rightarrow 0$.
[1X3.6 Description of the simple Lie algebras[0X
If [3XF[0m is a finite field, then, up to isomorphism, there is precisely one
simple Lie algebra with dimension 3, and another one with dimension 6; these
can be accessed by calling [3XNonSolvableLieAlgebra(F,[3,1])[0m and
[3XNonSolvableLieAlgebra(F,[6,2])[0m (see [3XNonSolvableLieAlgebra[0m for the details).
Over a field of characteristic 5, there is an additional simple Lie algebra
with dimension 5, namely [3XNonSolvableLieAlgebra(F,[5,3])[0m. These are the only
isomorphism types of simple Lie algebras over finite fields up to dimension
6.
In addition to the algebras above the package contains the simple Lie
algebras of dimension between 7 and 9 over [3XGF(2)[0m. These Lie algebras were
determined by [Vau06] and can be described as follows.
There are two isomorphism classes of 7-dimensional Lie algebras over [3XGF(2)[0m.
In a basis b1,...,b7 the non-trivial products in the first algebra are
[b1,b2]=b3, [b1,b3]=b4, [b1,b4]=b5, [b1,b5]=b6
[b1,b6]=b7, [b1,b7]=b1, [b2,b7]=b2, [b3,b6]=b2,
[b4,b5]=b2, [b4,b6]=b3, [b4,b7]=b4, [b6,b7]=b6;
and those in the second are
[b1,b2]=b3, [b1,b3]=b1+b4, [b1,b4]=b5, [b1,b5]=b6,
[b1,b6]=b7, [b2,b3]=b2, [b2,b5]=b2+b4, [b2,b6]=b5,
[b2,b7]=b1+b4, [b3,b4]=b2+b4, [b3,b5]=b3, [b3,b6]=b1+b4+b6,
[b3,b7]=b5, [b4,b7]=b6, [b5,b6]=b6, [b5,b7]=b7.
Over [3XGF(2)[0m there are two isomorphism types of simple Lie algebras with
dimension 8. In the basis b1,...,b8 the non-trivial products for the first
one are
[b1,b3]=b5, [b1,b4]=b6, [b1,b7]=b2, [b1,b8]=b1, [b2,b3]=b7, [b2,b4]=b5+b8,
[b2,b5]=b2, [b2,b6]=b1, [b2,b8]=b2, [b3,b6]=b4, [b3,b8]=b3, [b4,b5]=b4,
[b4,b7]=b3, [b4,b8]=b4, [b5,b6]=b6, [b5,b7]=b7, [b6,b7]=b8;
and for the second one they are
[b1,b2]=b3, [b1,b3]=b2+b5, [b1,b4]=b6, [b1,b5]=b2, [b1,b6]=b1+b4+b8,
[b1,b8]=b4, [b2,b3]=b4, [b2,b4]=b1, [b2,b5]=b6, [b2,b6]=b2+b7,
[b2,b7]=b2+b5, [b3,b4]=b2+b7, [b3,b5]=b1+b4+b8, [b3,b6]=b1, [b3,b7]=b2+b3,
[b3,b8]=b1, [b4,b5]=b3, [b4,b6]=b2+b4, [b4,b7]=b1+b4+b8, [b4,b8]=b3,
[b5,b6]=b1+b2+b5, [b5,b7]=b3, [b5,b8]=b2+b7, [b6,b7]=b4+b6, [b6,b8]=b2+b5,
[b7,b8]=b6.
The non-trivial products for the unique simple Lie algebra with dimension 9
over [3XGF(2)[0m are as follows:
[b1,b2]=b3, [b1,b3]=b5, [b1,b5]=b6, [b1,b6]=b7, [b1,b7]=b6+b9,
[b1,b9]=b2, [b2,b3]=b4, [b2,b4]=b6, [b2,b6]=b8, [b2,b8]=b6+b9,
[b2,b9]=b1, [b3,b4]=b7, [b3,b5]=b8, [b3,b7]=b1+b8, [b3,b8]=b2+b7,
[b4,b5]=b6+b9, [b4,b6]=b2+b7, [b4,b7]=b3+b6+b9, [b4,b9]=b5,
[b5,b6]=b1+b8, [b5,b8]=b3+b6+b9, [b5,b9]=b4, [b6,b7]=b1+b4+b8,
[b6,b8]=b2+b5+b7, [b7,b8]=b3+b9, [b7,b9]=b8, [b8,b9]=b7.
[1X3.7 Description of the solvable and nilpotent Lie algebras[0X
In this section we list the multiplication tables of the nilpotent and
solvable Lie algebras contained in the package. Some parametric classes
contain isomorphic Lie algebras, for different values of the parameters. For
exact descriptions of these isomorphisms we refer to [dG05], [dG07]. In
dimension 2 there are just two classes of solvable Lie algebras:
-- L_2^1: The Abelian Lie algebra.
-- L_2^2: [x_2,x_1]=x_1.
We have the following solvable Lie algebras of dimension 3:
-- L_3^1 The Abelian Lie algebra.
-- L_3^2 [x_3,x_1]=x_1, [x_3,x_2]=x_2.
-- L_3^3(a) [x_3,x_1]=x_2, [x_3,x_2]=ax_1+x_2.
-- L_3^4(a) [x_3,x_1]=x_2, [x_3,x_2]=ax_1.
And the following solvable Lie algebras of dimension 4:
-- L_4^1 The Abelian Lie algebra.
-- L_4^2 [x_4,x_1]=x_1, [x_4,x_2]=x_2, [x_4,x_3]=x_3.
-- L_4^3(a) [x_4,x_1]=x_1, [x_4,x_2]=x_3, [x_4,x_3]=-ax_2 +(a+1)x_3 .
-- L_4^4 [x_4,x_2]=x_3, [x_4,x_3]= x_3 .
-- L_4^5 [x_4,x_2]=x_3 .
-- L_4^6(a,b) [x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2+x_3 .
-- L_4^7(a,b) [x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2.
-- L_4^8 [x_1,x_2]=x_2, [x_3,x_4]=x_4 .
-- L_4^9(a) [x_4,x_1] = x_1+ax_2, [x_4,x_2]=x_1, [x_3,x_1]=x_1,
[x_3,x_2]=x_2 .
-- L_4^10(a) [x_4,x_1] = x_2, [x_4,x_2]=ax_1, [x_3,x_1]=x_1,
[x_3,x_2]=x_2 Condition on F: the characteristic of F is 2.
-- L_4^11(a,b) [x_4,x_1] = x_1, [x_4,x_2] = bx_2, [x_4,x_3]=(1+b)x_3,
[x_3,x_1]=x_2, [x_3,x_2]=ax_1. Condition on F: the characteristic of F
is 2.
-- L_4^12 [x_4,x_1] = x_1, [x_4,x_2]=2x_2, [x_4,x_3] = x_3, [x_3,x_1]=x_2
.
-- L_4^13(a) [x_4,x_1] = x_1+ax_3, [x_4,x_2]=x_2, [x_4,x_3] = x_1,
[x_3,x_1]=x_2 .
-- L_4^14(a) [x_4,x_1] = ax_3, [x_4,x_3]=x_1, [x_3,x_1]=x_2 .
Nilpotent of dimension 5:
-- N_5,1 Abelian.
-- N_5,2 [x_1,x_2]=x_3 .
-- N_5,3 [x_1,x_2]=x_3, [x_1,x_3]=x_4 .
-- N_5,4 [x_1,x_2] = x_5, [x_3,x_4]=x_5 .
-- N_5,5 [x_1,x_2]=x_3, [x_1,x_3]= x_5, [x_2,x_4] = x_5 .
-- N_5,6 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5 .
-- N_5,7 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5 .
-- N_5,8 [x_1,x_2]=x_4, [x_1,x_3]=x_5 .
-- N_5,9 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_2,x_3]=x_5 .
We get nine 6-dimensional nilpotent Lie algebras denoted N_6,k for k=1,...,9
that are the direct sum of N_5,k and a 1-dimensional abelian ideal.
Subsequently we get the following Lie algebras.
-- N_6,10 [x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_4,x_5]=x_6.
-- N_6,11 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_3]=x_6,
[x_2,x_5]=x_6 .
-- N_6,12 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_5]=x_6 .
-- N_6,13 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_2,x_4]=x_5, [x_1,x_5]=x_6,
[x_3,x_4]=x_6 .
-- N_6,14 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5,
[x_2,x_5]=x_6,[x_3,x_4]=-x_6 .
-- N_6,15 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5,
[x_1,x_5]=x_6,[x_2,x_4]=x_6 .
-- N_6,16 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_5]=x_6,
[x_3,x_4]=-x_6 .
-- N_6,17 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6,
[x_2,x_3]= x_6 .
-- N_6,18 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6 .
-- N_6,19(a) [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_4]=x_6, [x_3,x_5]=a x_6
.
-- N_6,20 [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_6 .
-- N_6,21(a) [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_2,x_3]=x_5, [x_1,x_4]=x_6,
[x_2,x_5]= a x_6 .
-- N_6,22(a) [x_1,x_2]=x_5, [x_1,x_3]=x_6, [x_2,x_4]= a x_6,
[x_3,x_4]=x_5 .
-- N_6,23 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6, [x_2,x_4]= x_5 .
-- N_6,24(a) [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=a x_6,
[x_2,x_3]=x_6, [x_2,x_4]= x_5 .
-- N_6,25 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6 .
-- N_6,26 [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6 .
distrib
>
Mandriva
>
2010.0
>
i586
>
media
>
contrib-release
>
by-pkgid
>
5e1854624d3bc613bdd0dd13d1ef9ac7
>
files
>
2179
